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# Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups

Published:2018-04-04

• Yanchang Han,
School of Mathematical Sciences, South China Normal University , Guangzhou, 510631, China
• Yongsheng Han,
Department of Mathematics, Auburn University , Auburn, AL 36849, USA
• Ji Li,
Department of Mathematics, Macquarie University , Sydney NSW 2109, Australia
• Chaoqiang Tan,
Department of Mathematics, Shantou University , Shantou, Guangdong 515041, China
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## Abstract

The Marcinkiewicz multipliers are $L^{p}$ bounded for $1\lt p\lt \infty$ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (Müller, Ricci and Stein). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while there is \emph{no} two parameter group of \emph{automorphic} dilations on $\mathbb{H} ^{n}$. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ in the sense `intermediate' between the classical Lipschitz space on the Heisenberg group $\mathbb{H}^{n}$ and the product Lipschitz space on $\mathbb{C}^{n}\times \mathbb{R}$. We characterize this flag Lipschitz space via the Littlewood-Paley theory and prove that flag singular integral operators, which include the Marcinkiewicz multipliers, are bounded on these flag Lipschitz spaces.
 Keywords: Heisenberg group, Marcinkiewicz multiplier, Flag singular integral, Flag Lipschitz space, reproducing formula, Discrete Littlewood--Paley analysis
 MSC Classifications: 42B25 - Maximal functions, Littlewood-Paley theory 42B35 - Function spaces arising in harmonic analysis 43A17 - Analysis on ordered groups, $H^p$-theory

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